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http://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem states that independence of "The strengthened finite Ramsey theorem" from PA is proved by implying consistency of PA. But how do we know that it's actually true for the Model that's Natural Numbers. How was it even proved for Natural Numbers? Under what logical system is it proved, if not for Peano's Axioms? And how do we know that such a logical system on which it's proved actually models Natural Numbers?

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No, that article doesn't state that. That doesn't even make sense. –  Chris Eagle Oct 21 '12 at 11:44
ha, i meant the "independence" of the theorem is proved by implying the consistency of PA. Editing the question now. Thanks! –  rajeshsr Oct 21 '12 at 13:46

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One of the common frameworks for mathematics is the axiomatic set theory ZFC. This system proves the consistency of second-order PA. In particular, we can show that the finite von Neumann ordinals form a model of this system.

In particular every model of second-order PA is a model of first-order PA.

The strengthened Ramsey theorem can be proved in second-order PA, as stated in the Wikipedia page linked in the question. Therefore ZFC proves that this theorem is true for the natural numbers.

As discussed in this question when we say that a theorem is "true for the natural numbers" we mean that it holds for the standard model of PA (which is the model of second-order PA).

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Thanks! Second-order arithmetics is the key-word i am looking for, i guess. Starting to read about it. Just some general questions, if it's possible to answer without some heavy machinery: Can you give some intuition as to how second-order arithmetics has only one model and what "axiomatic power" it has more compared to first-order arithmetics to prove Ramsey theorem? Is there some good resource anyone can recommend for understanding specifically about "first-order" vs "second-order" arithmetics? –  rajeshsr Oct 21 '12 at 13:58
@rajeshsr: I'm sorry. I missed your comment. I think that asking for resources in order to understand the difference would be a reasonable question on its own. You may want to start with the following: math.stackexchange.com/questions/23799 math.stackexchange.com/questions/141759 math.stackexchange.com/questions/123566 math.stackexchange.com/questions/79536 math.stackexchange.com/questions/123566 –  Asaf Karagila Oct 22 '12 at 14:17

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